Spacelike intersection curve of three spacelike hypersurfaces in E4/1

U N I V E R S I T A T I S M A R I A E C U R I E - S K Ł O D O W S K A L U B L I N – P O L O N I A

VOL. LXVII, NO. 1, 2013 SECTIO A 23–33

B. UYAR D ¨ULD ¨UL and M. C¸ ALIS¸KAN

Spacelike intersection curve of three spacelike hypersurfaces in E

Abstract. In this paper, we compute the Frenet vectors and the curvatures of the spacelike intersection curve of three spacelike hypersurfaces given by their parametric equations in four-dimensional Minkowski space E⁴₁.

1. Introduction. The surface-surface intersection(SSI) is one of the basic problems in computational geometry. The main purpose here is to deter- mine the intersection curve between the surfaces and to get information about the geometrical properties of the curve. Since the surfaces are mostly given by their parametric or implicit equations, three cases are valid for the SSI problems: parametric-parametric, implicit-implicit and parametric- implicit.

There are two types of SSI problems: transversal or tangential. The inter- section at the intersecting points is called transversal if the normal vectors of the surfaces are linearly independent, and is called tangential if the nor- mal vectors of the surfaces are linearly dependent. The tangent vector of the intersection curve can be obtained easily by the vector product of the normal vectors of the surfaces in transversal intersection problems. There- fore, so many studies have recently been done about this type of problems.

Hartmann [6], provides formulas for computing the curvatures of the in- tersection curves for all types of intersection problems in three-dimensional Euclidean space. Willmore [11], and using the implicit function theorem

2010 Mathematics Subject Classification. 53A35, 53A04, 53A05.

Key words and phrases. Intersection curve, hypersurface.

Alessio [1], give the methods to compute the unit tangent, the unit principal normal, the unit binormal vectors, and then the curvature and the torsion of the transversal intersection curve of two implicit surfaces. Goldman [5], provides formulas for computing the curvature and the torsion of intersec- tion curve of two implicit surfaces, using the classical curvature formulas in Differential Geometry. Ye and Maekawa [12], present algorithms to com- pute the Frenet vectors and curvatures of the intersection curve for all three types of transversal and tangential intersections. Alessio and Guadalupe [2], give formulas for computing the local properties of a transversal intersection curve of two spacelike surfaces in the Lorentz–Minkowski 3-space. Using the implicit function theorem, Alessio [3], presents algorithms for computing the differential geometry properties of intersection curves of three implicit sur- faces in R⁴. D¨uld¨ul [4], gives methods for computing the Frenet apparatus of the transversal intersection curve of three parametric hypersurfaces in four-dimensional Euclidean space.

In this paper, we find the tangent, the principal normal, the first and sec- ond binormal vectors and the first, second and third curvatures of the space- like intersection curve of three parametric spacelike hypersurfaces which are intersecting transversally in E₁⁴.

2. Preliminaries. The Minkowski space E₁⁴ is the real vector space R⁴ provided with the (standard flat) metric given by

g = −dx²₁+ dx²₂+ dx²₃+ dx²₄,

where (x₁, x₂, x₃, x₄) is a rectangular coordinate system of E₁⁴. A vector u ∈ E₁⁴ is called a spacelike, a timelike, and a null (lightlike) vector if g(u, u) > 0 or u = 0, g(u, u) < 0, and g(u, u) = 0 for u 6= 0, respectively, [8]. The norm of a vector u is defined by ||u|| =p|g(u, u)|. Two vectors u and v are said to be orthogonal if g(u, v) = 0. A vector u satisfying g(u, u) = ±1 is called a unit vector. For an arbitrary curve α = α(s) in E₁⁴, if all of its velocity vectors α⁰(s) are spacelike, timelike or null vectors, then the curve is called a spacelike, a timelike or a null curve, respectively, [8].

A hypersurface in E₁⁴ is called a timelike (spacelike) hypersurface if the induced metric on the hypersurface is a Lorentz (positive definite Riemann- ian) metric. The normal vector on the timelike (spacelike) hypersurface is a spacelike (timelike) vector.

The ternary product of the vectors u = P4

i=1u_ie_i, v = P4

i=1v_ie_i, and w =P4

i=1wiei is defined by

u ⊗ v ⊗ w = −        

−e₁ e2 e3 e4

u1 u2 u3 u4

v₁ v₂ v₃ v₄ w1 w2 w3 w4

        ,

where {e₁, e₂, e₃, e₄} is the standard basis of four-dimensional Minkowski space E₁⁴.

For the vectors e₁, e2, e3, e4 the following equations are satisfied:

e₁⊗ e₂⊗ e₃= e₄, e₂⊗ e₃⊗ e₄= e₁, e₃⊗ e₄⊗ e₁= e₂, e₄⊗ e₁⊗ e₂= −e₃, [13].

From the definition of ternary product, we get g(u, u ⊗ v ⊗ w) = g(v, u ⊗ v ⊗ w) = g(w, u ⊗ v ⊗ w) = 0; that is, the vector u ⊗ v ⊗ w is orthogonal to u, v and w. The triple vector product of vectors u, v and w in E₁³ is defined by

(u × v) × w =    

u v

g(u, w) g(v, w)    

= g(v, w)u − g(u, w)v,

a linear combination of vectors u and v, [7, 10]. In an analogous manner in E₁⁴ we can express the quintuple vector product of vectors u, v, w, x, and y as

(u ⊗ v ⊗ w) ⊗ x ⊗ y =      

u v w

g(u, x) g(v, x) g(w, x) g(u, y) g(v, y) g(w, y)       ,

a linear combination of vectors u, v and w (see [10] for the Euclidean case).

We may also write

g(u ⊗ v ⊗ w, x ⊗ y ⊗ z) = −      

g(u, x) g(v, x) g(w, x) g(u, y) g(v, y) g(w, y) g(u, z) g(v, z) g(w, z)

      .

Let {t(s), n(s), b₁(s), b₂(s)} be the moving Frenet frame along the curve α(s) in the Minkowski 4-space E₁⁴. Then t(s), n(s), b₁(s), b₂(s) denote the tangent, the principal normal, the first binormal, and the second binormal vector fields, respectively.

Let α(s) be a spacelike curve with arc length parameter s in E⁴₁. Then t(s) = α⁰(s) is a spacelike unit vector, i.e., ||α⁰(s)|| = 1. Therefore g(α⁰(s), α⁰(s)) = 1 and g(α⁰(s), α⁰⁰(s)) = 0. Depending on the vector α⁰⁰(s) we investigate the following cases [9].

Case 1: n is spacelike:

Case 1.1: The second binormal vector b₂ is the unique timelike vector in the Frenet frame {t, n, b₁, b₂}. Then the Frenet formulas are











t⁰ = k1n

n⁰ = −k₁t + k₂b₁ b⁰₁ = −k2n + k3b2

b⁰₂ = k₃b₁,

where k_i’s (i = 1, 2, 3) are the ith curvature functions of the curve α.

Case 1.2: The first binormal vector b₁ is the unique timelike vector in the tetrad {t, n, b₁, b₂}. In this case, the Frenet formulas are given by











t⁰ = k₁n

n⁰ = −k1t + k2b1

b⁰₁ = k₂n + k₃b₂ b⁰₂ = −k3b1. Case 2: n is timelike:

The Frenet formulas have the form











t⁰= k1n

n⁰ = k₁t + k₂b₁ b⁰₁= k₂n + k₃b₂ b⁰₂= −k3b1.

From elementary differential geometry, we know that α⁰(s) = t(s). Also, the second derivative α⁰⁰(s) is equal to k₁n in all above cases. The third derivative α⁰⁰⁰(s) can be obtained as α⁰⁰⁰ = k⁰₁n + k1n⁰ by differentiating the equation α⁰⁰= k₁n. If we replace n⁰ using the Frenet formulas, we get

Case 1:

(2.1) α⁰⁰⁰= −k²₁t + k₁⁰n + k1k2b1. Case 2:

(2.2) α⁰⁰⁰ = k²₁t + k₁⁰n + k1k2b1.

Then the second curvature k₂ of the curve α can be obtained from Eq. (2.1) and Eq. (2.2) as

Case 1.1 and Case 2: k₂ = ^g(α000_k^,b1)

1 .

Case 1.2: k₂ = −^g(α000_k^,b1)

1 .

Now let us find the fourth derivative α⁽⁴⁾(s) similar to the third derivative α⁰⁰⁰(s):

Case 1.1: α⁽⁴⁾ = −3k₁k⁰₁t + (−k³₁+ k₁⁰⁰− k₁k₂²)n + (2k⁰₁k₂+ k₁k⁰₂)b₁ + k1k2k3b2

Case 1.2: α⁽⁴⁾ = −3k₁k⁰₁t + (−k³₁+ k₁⁰⁰+ k₁k₂²)n + (2k⁰₁k₂+ k₁k⁰₂)b₁ + k₁k₂k₃b₂

Case 2: α⁽⁴⁾ = 3k₁k₁⁰t+(k₁³+ k₁⁰⁰+ k₁k²₂)n+(2k⁰₁k₂+ k₁k⁰₂)b₁+k₁k₂k₃b₂. Using the above equations, the third curvature of α can be found by k₃ = −^g(^α(4),b2)

k1k2 for Case 1.1 and k₃ = ^g(^α(4),b2)

k1k2 for Case 1.2 and Case 2.

3. The curvatures of spacelike intersection curve. Let Mi(i = 1, 2, 3) be three spacelike hypersurfaces with parametric equations Xⁱ = Xⁱ(ui, vi, wi). Let us assume these hypersurfaces intersect transversally at an intersection point α(s₀) = P on the spacelike intersection curve α(s).

Then, the unit normal vectors N_i of these hypersurfaces are timelike vec- tors and they can be given by

N_i= X_uⁱ_i⊗ X_vⁱ

i⊗ X_wⁱ

i

||X_uⁱ

i⊗ X_vⁱ

i⊗ X_wⁱ

i||.

Since the intersection curve α is a spacelike curve, the unit tangent vector t of α is a spacelike vector and can be obtained by the ternary product of the normal vectors at P :

t = N1⊗ N₂⊗ N₃

||N₁⊗ N₂⊗ N₃||.

Since the intersection is transversal, the normal vectors N_i are linearly independent at the intersection point, i.e., N₁⊗ N₂⊗ N₃ 6= 0.

3.1. First Curvature. From the Frenet equations, we know k1 = ||t⁰||.

Then we must find the vector t⁰ for calculating the first curvature k₁. Since t⁰ is orthogonal to t, we can write

(3.1) α⁰⁰ = t⁰ = a₁N₁+ a₂N₂+ a₃N₃, a_i ∈R^.

Now let us determine the scalars a_i to find α⁰⁰. If we take the dot product of both hand sides of (3.1) with N_i, then we get

(3.2) g(N₁, N_i)a₁+ g(N₂, N_i)a₂+ g(N₃, N_i)a₃= K_nⁱ,

where K_nⁱ = g(t⁰, Ni), (i = 1, 2, 3). The coefficients determinant of the linear system (3.2) is ∆ = −||N₁⊗ N₂⊗ N₃||². Since ∆ is different from zero, solving the coefficients from linear system (3.2) yields (as in [4])

ai = 1

∆{− sinh²θjkK_nⁱ + bijK_n^j + bikK_n^k}, i, j, k = 1, 2, 3 (cyclic), where θ_ij is the angle between the timelike unit normal vectors N_i and N_j. Also, if N₁, N2 and N₃ are in the same timecone of E₁⁴, then

bij = cosh θ_ikcosh θ_jk− cosh θ_ij. If N₁, N₂ and N₃ are not in the same timecone of E₁⁴, then

bij = cosh θikcosh θjk+ cosh θij.

If N₁ and N₂ are in the same timecone but N₃ is in the different timecone, then

bij = (−1)^i+kcosh θ_ikcosh θ_jk − cosh θ_ij.

If N₁ and N₃ are in the same timecone but N₂ is in the different timecone, then

b_ij = (−1)^i+jcosh θ_ikcosh θ_jk− cosh θ_ij.

If N₂ and N₃ are in the same timecone but N₁ is in the different timecone, then

b_ij = (−1)^j+kcosh θ_ikcosh θ_jk− cosh θ_ij.

The normal curvatures K_nⁱ needed to find the scalars a_i are calculated as expressed in [4]. Then the first curvature k₁ of the intersection curve α at P is given as follows:

If N₁, N₂ and N₃ are in the same timecone, then k²₁ =

a²₁+ a²₂+ a²₃+ 2a1a2cosh θ12+ 2a1a3cosh θ13+ 2a2a3cosh θ23

 . If N₁, N₂ and N₃ are not in the same timecone, then

k²₁ =

a²₁+ a²₂+ a²₃− 2a₁a2cosh θ12− 2a₁a3cosh θ13− 2a₂a3cosh θ23

 . If N₁ and N₂ are in the same timecone but N₃ is in the different timecone, then

k²₁ =

a²₁+ a²₂+ a²₃+ 2a₁a₂cosh θ₁₂− 2a₁a₃cosh θ₁₃− 2a₂a₃cosh θ₂₃ . If N₁ and N₃ are in the same timecone but N₂ is in the different timecone, then

k²₁ =

a²₁+ a²₂+ a²₃− 2a₁a2cosh θ12+ 2a1a3cosh θ13− 2a₂a3cosh θ23

 . If N₂ and N₃ are in the same timecone but N₁ is in the different timecone, then

k²₁ =

a²₁+ a²₂+ a²₃− 2a₁a₂cosh θ₁₂− 2a₁a₃cosh θ₁₃+ 2a₂a₃cosh θ₂₃ . 3.2. Second Curvature. Now, let us find the second curvature of the intersection curve α at P . Since the timelike unit normal vectors N_i are orthogonal to t, the terms k⁰₁n + k₁k₂b₁ in Eq. (2.1) and Eq. (2.2) can be replaced by c₁N₁+ c₂N₂+ c₃N₃. Thus

Case 1: α⁰⁰⁰= −k²₁t + c1N1+ c2N2+ c3N3, Case 2: α⁰⁰⁰= k₁²t + c₁N₁+ c₂N₂+ c₃N₃.

If the dot products of both hand sides of above equations are taken with N_i, we have a linear equation system similar to (3.2). Solving this system yields

(3.3) c_i= 1

∆{− sinh²θ_jkµ_i+ b_ijµ_j + b_ikµ_k}, i, j, k = 1, 2, 3 (cyclic), where µ_i= g(α⁰⁰⁰, N_i). The scalars µ_iin Eq. (3.3) are computed as explained in [4]. Then, the second curvature k₂ of the intersection curve α at P is given by

Case 1.1 and Case 2: k₂ = ^g(α000_k^,b1)

1 ,

Case 1.2: k₂ = −^g(α000_k^,b1)

1 .

3.3. Third Curvature. Now, let us compute the third curvature k3 of the curve α at P . Similar to the second and third derivatives of α, we may write

Case 1: α⁽⁴⁾= −3k₁k₁⁰t + d₁N₁+ d₂N₂+ d₃N₃,

Case 2: α⁽⁴⁾= 3k₁k⁰₁t + d₁N₁+ d₂N₂+ d₃N₃, where d_i= 1

∆{− sinh²θ_jkξ_i+ b_ijξ_j+ b_ikξ_k}, i, j, k = 1, 2, 3 (cyclic), and ξ_i = g(α⁽⁴⁾, Ni). Obtaining the scalars ξi is mentioned in [4].

Thus, the third curvature of the intersection curve is found as Case 1.1: k₃ = −^g(α_k^(4),b2)

1k2 ,

Case 1.2 and Case 2: k₃ = ^g(α_k^(4),b2)

1k2 . 4. Examples.

4.1. Example 1. Let us consider the spacelike hypersurfaces M₁: X¹(u₁, v₁, w₁) =



cosh u₁+1

2, sinh u₁, v₁, w₁

 ,

M2: X²(u2, v2, w2) = (cosh u2, sinh u2cos v2, sinh u2sin v2, w2) , M3: X³(u3, v3, w3)

=

 1

√

2cosh u3+1 2, 1

√

2sinh u3sin v3+1

2, w3, 1

√

2sinh u3cos v3



in the Minkowski 4-space, where u₂6= 0, u₃ 6= 0. Let us compute the Frenet apparatus of the spacelike intersection curve α at the intersection point

P = X¹ 0,

√5 2 ,1

2

!

= X² ln 3 +√ 5 2

! ,π

2,1 2

!

= X³ ln√

2 + 1

 ,7π

4 ,

√ 5 2

!

= 3

2, 0,

√ 5 2 ,1

2

! .

At this point, the timelike unit normal vectors of these hypersurfaces are, respectively,

N₁ = (1, 0, 0, 0), N₂= 3 2, 0,

√5 2 , 0

!

, N₃ =

√ 2, − 1

√2, 0, 1

√2



and the spacelike unit tangent vector of α is t =

 0, 1

√ 2, 0, 1

√ 2

 .

Since g(N₁, N2), g(N1, N3) and g(N2, N3) are smaller than zero, the time- like unit normal vectors N_i, 1 ≤ i ≤ 3, are in the same timecone. Then, cosh θ12 = ³₂, cosh θ₁₃ =√

2, cosh θ23 = ^√3

2 and so b₁₂ = ³₂, b₁₃ = ⁵

2√ 2 and b₂₃= 0.

The non-vanishing first fundamental form coefficients of these hypersur- faces at P are

g₁₁¹ = g₂₂¹ = g₃₃¹ = g₁₁² = g₃₃² = g³₃₃= 1, g²₂₂= 5

4, g₁₁³ = g₂₂³ = 1 2 and the non-vanishing second fundamental form coefficients of these hyper- surfaces at P are

h¹₁₁= h²₁₁= −1, h²₂₂= −5

4, h³₁₁= h³₂₂= − 1

√2.

As expressed in [4], the values u⁰_i, v_i⁰ and w_i⁰ are found as:

u⁰₁ = 1

√2, v⁰₁= 0, w⁰₁ = 1

√2, u⁰₂ = 0, v⁰₂= −

√2

√5, w⁰₂ = 1

√2, u⁰₃ = 0, v⁰₃=√

2, w⁰₃ = 0.

Hence, for the normal curvatures K_nⁱ we find K_n¹ = K_n² = −¹₂, K_n³ = −√ 2.

Thus, we have a1= 6

5, a2= 1

5, a3= − 1

√2, k1=

√

√3 10, α⁰⁰= 1

2,1 2, 1

2√ 5, −1

2



, n =

√

√5 6,

√

√5 6, 1

√ 6, −

√

√5 6

! ,

where n is a spacelike vector. Also, the values u⁰⁰_i, v_i⁰⁰ and w⁰⁰_i are calculated as:

u⁰⁰₁ = 1

2, v₁⁰⁰= 1 2√

5, w₁⁰⁰= −1 2, u⁰⁰₂ = 1

√

5, v₂⁰⁰= − 1

√

5, w₂⁰⁰= −1 2, u⁰⁰₃ = 1

√

2, v₃⁰⁰= 0, w₃⁰⁰= 1 2√

5. Using above, we get µ₁= µ₂= − ³

2√

2 and µ₃ = 0. So, the coefficients c_i are found as c₁ = − ¹²

5√

2, c₂= ³

5√

2, c₃ = ³₂. Then, we get α⁰⁰⁰ = 3

2√ 2, − 9

5√ 2, 3√

5 10√

2, 6 5√

2

! , α⁰⊗ α⁰⁰⊗ α⁰⁰⁰=

 3 2√

5, 0,3 2, 0



, ||α⁰⊗ α⁰⁰⊗ α⁰⁰⁰|| = 3

√ 5,

b2= α⁰⊗ α⁰⁰⊗ α⁰⁰⁰

||α⁰⊗ α⁰⁰⊗ α⁰⁰⁰|| = 1 2, 0,

√ 5 2 , 0

! ,

b₁= b₂⊗ α⁰⊗ α⁰⁰

||b₂⊗ α⁰⊗ α⁰⁰|| = 5 2√

3, 1

√3,

√5 2√

3, − 1

√3

! .

Note that b₂ is a spacelike vector and b₁ is the unique timelike vector in the tetrad {t, n, b₁, b2}. Because of this, Case 1.2 is valid. Then the second curvature k₂ of α is found as k₂ = 2√

5. Also, k⁰₁= g(α⁰⁰⁰, n) = −⁷

√15 10 . For the values u⁰⁰⁰_i , v⁰⁰⁰_i and w_i⁰⁰⁰, we obtain

u⁰⁰⁰₁ = − 23 10√

2, v⁰⁰⁰₁ = 3√ 5 10√

2, w⁰⁰⁰₁ = 6 5√

2, u⁰⁰⁰₂ = 3

√

10, v⁰⁰⁰₂ = 32 5√

10, w⁰⁰⁰₂ = 6 5√

2, u⁰⁰⁰₃ = 3

2, v⁰⁰⁰₃ = − 13 5√

2, w⁰⁰⁰₃ = 3 2√

10. Hence, we have

ξ1 = 18

5 , ξ2= 69

20, ξ3 = − 21 10√

2 and using these values

d₁ = 201

25 , d₂ = −39

25, d₃= − 93 10√

2 are found. So, we obtain

α⁽⁴⁾ =



−18 5 ,38

5 , − 39 10√

5, −3 2

 , k₃ = − 3

20√ 6.

4.2. Example 2. Let M₁, M₂, and M₃be the spacelike hypersurfaces given by, respectively,

X¹(u₁, v₁, w₁) =



cosh u₁, sinh u₁+ 1 2, v₁, w₁

 , X²(u2, v2, w2) = 1

2cosh u2,1

2sinh u2cos v2, w2,1

2sinh u2sin v2

 , X³(u3, v3, w3) = 1

2cosh u3, w3,1

2sinh u3sin v3−1 2,1

2sinh u3cos v3

 ,

where u₂ 6= 0, u₃ 6= 0. The Frenet vectors and the curvatures of the spacelike intersection curve of these surfaces at the intersection point

P = X¹

 0, 0, 1

√2



= X² ln

2 +√ 3

, arctan√ 2, 0

= X³

 ln

 2 +√

3



, arctan 1

√ 2 ,1

2



=

 1,1

2, 0, 1

√ 2



are found as:

t = 0, −

√

√2 5, −

√

√2 5, 1

√5

!

, n = 5

√15, − 1

√15, − 1

√15, −2√

√ 2 15

! ,

b1= −

√2

√ 3, 1

√ 6, 1

√ 6, 2

√ 3

!

, b2=

 0, − 1

√ 2, 1

√ 2, 0

 ,

k₁ = 2√ 15

25 , k₂= 2, k₃ = 0.

Here, the timelike unit normal vectors N₂and N₃ are in the same timecone but N₁ is in the different timecone. Also, n is the unique timelike vector in the Frenet frame {t, n, b₁, b2} and for this reason, Case 2 is valid.

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B. Uyar D¨uld¨ul M. C¸ alı¸skan

Department of Mathematics Education Department of Mathematics

Education Faculty Science Faculty

Yıldız Technical University Gazi University

˙Istanbul Ankara

Turkey Turkey

e-mail: buduldul@yildiz.edu.tr e-mail: mustafacaliskan@gazi.edu.tr Received February 2, 2012